2nd in command to sauron

[Eurytus]

The Mouth of Sauron was the Lt. of Barad-dur whether the Witch-King was live or not because the Witch-King did not reside at Barad-dur. Khamul *was* the highest power at Dol Guldur - see Gwaihir's quote above.

I was not discussing whether the Mouth was the lieutenant of Barad Dur, that much is obvious. I was discussing whether he was now the lieutenant of the whole of Sauron's forces which his statement suggests he is.

As to Khamul. He was the highest power at Dol Guldur when there were no military activities going on. None of the Nazgul save the Witchking were trusted with commanding armies.
Therefore we have this plus the statement of the Mouth himself to suggest that Sauron would have made the Mouth his 2nd in command.
Any of the Nazgul would have been wholely unsuited to the role in any case as the job of Sauron's lieutenant would have been to control and treat with the client kingdoms beyond the Anduin. Given the terror that went before them a Nazgul would have been unsuitable.